Cyclic extensions of $K(\sqrt {1})/K$
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 by Jón Kr. Arason, Burton Fein, Murray Schacher and Jack Sonn PDF
 Trans. Amer. Math. Soc. 313 (1989), 843851 Request permission
Abstract:
In this paper the height ${\text {ht}}(L/K)$ of a cyclic $2$extension of a field $K$ of characteristic $\ne 2$ is studied. Here ${\text {ht}}(L/K) \geq n$ means that there is a cyclic extension $E$ of $K,E \supset L$, with $[E:L] = {2^n}$. Necessary and sufficient conditions are given for ${\text {ht}}(L/K) \geq n$ provided $K(\sqrt {  1})$ contains a primitive ${2^n}$th root of unity. Primary emphasis is placed on the case $L = K(\sqrt {  1})$. Suppose ${\text {ht}}(K(\sqrt {  1})/K) \geq 1$. It is shown that ${\text {ht}}(K(\sqrt {  1})/K) \geq 2$ and if $K$ is a number field then ${\text {ht}}(K(\sqrt {  1})/K) \geq n$ for all $n$. For each $n \geq 2$ an example is given of a field $K$ such that ${\text {ht}}(K(\sqrt {  1})/K) \geq n$ but ${\text {ht}}(K(\sqrt {  1})/K) \ngeq n + 1$.References

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Additional Information
 © Copyright 1989 American Mathematical Society
 Journal: Trans. Amer. Math. Soc. 313 (1989), 843851
 MSC: Primary 12F10; Secondary 11R20
 DOI: https://doi.org/10.1090/S00029947198909296652
 MathSciNet review: 929665